Publication: On Hilbert 17th problem and real nullstellensatz for global analytic functions
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1985
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Springer
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The author proves a Nullstellensatz for the ring of real analytic functions on a compact analytic manifold. The main results are the following. Theorem 1: Let X be a compact irreducible analytic set of a real analytic manifold M and f:X→R a nonnegative analytic function. Then f is a sum of squares of meromorphic functions. Theorem 2: Let I be a finitely generated ideal of O(M) with Z(I) compact. Then IZ(I)=I√R, where Z(I) is the zero set of I, IZ(I) the ideal of functions (in O(M)) vanishing on I, and I√R the real radical of I (i.e. the set of functions f in O(M) such that there exist g1,⋯,gk and an integer p with f2p + g2 1 + ⋯ +g2k ∈ I). Corollary: Let I be as in Theorem 2. Then IZ(I)=I if and only if I is real (i.e. I=I √ R). The proofs are based on results about extension of orders.
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Abhyankar, S.: On the valuation centered in a local domain, Am. J. Math. 78, 321-348 (1956)
Becket, E.: Valuations and real places in the theory of formally real fields, in Lecture Notes in Math. 959. Berlin Heidelberg New York: Springer 1982
Bochnak, J., Efroymson, G.: Real algebraic geometry and the 17th Hilbert Problem. Math. Ann. 251, 213-241 (1980)
Bochnak, J., Kucharz, W., Shiota, M.: On equivalence of ideals of real global analytic functions and the 17th Hilbert Problem. Invent. Math. 63, 403-421 (1981)
Bochnak, J., Risler, J.J.: Sur le théoréme des zéros pour les variétés analytiques réelles de dimension 2, Ann. Sci. Ec. Norm. Super. 8, 353-364 (1975)
Bourbaki, N.: Commutative Algebra. Paris: Hermann 1972
Bruhat, F., Whitney, H.: Quelques propriétés fondamentales des ensembles analytiques réels, Comment. Math. Helv. 33, 132-160 (1959)
Brumfiel, G.W.: Partially ordered rings and semialgebraic geometry. London Math. S. 37. Cambridge: University Press 1979
Brumfiel, G.W.: Real valuation rings and ideals, in Lecture Notes in Math. 959. Berlin Heidelberg New York: Springer 1982
Cartan, H.: Variétés analytiques réelles et variétés analytiques complexes. Bull. Soc. Math. France 85, 77-99 (1957)
Grothendieck, A., Dieudonné, J.: Elements de Geométrie Algebrique. Publ. Math. I.H.E.S. 24 (1965)
Frisch, J.: Points de platitude d'un morphisme d'espaces analytiques complexes, Invent. Math. 4, 118-138 (1967)
Jacobson, N.: Lectures in Abstract Algebra, III, Graduate Texts in Math. 32. Berlin Heidelberg New York: Springer
Jaworski, P.: Positive definite analytic functions and vector bundles. Bull. Acad. Pol. Sci., XXX, nº 11-12, 501-506 (1982)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. I. Interscience Tracts in Pure and Appl. Math. 15. New York: J. Wiley 1963
Lassalle, G.: Sur le théoréme des zéros differentiables, in Lecture Notes in Math. 535. Berlin Heidelberg New York: Springer 1975
Lam, T.Y.: An introduction to real algebra. Sexta Escuela Latinoamericana de Mat. Oaxtepec, 1982
Lam, T.Y.: Orderings, valuations and quadratic forms. Conf. Board Math. Sciences, 52, AMS 1983
Matsumura, H.: Commutative Algebra, 2d edition. Amsterdam: W.A. Benjamin Co. 1980
Risler, JJ.: Le théorème des zéros en géométries algébrique et analytique réelles. Bull. Soc. Math. Fr. 104, 113-127 (1976)
Ruiz, J.M.: Central orderings in fields of real meromorphic function germs. Manuscr. Math. 46, 193-214 (1984)
Ruiz, J.M.: Local theory of total orderings in excellent domains. Preprint 1984
Ruiz, J.M.: A quantitative theorem on extensions of orderings under completion. Preprint 1984